Polar actions on Damek-Ricci spaces

TL;DR

This paper investigates polar isometric actions on Damek-Ricci spaces, establishing criteria, providing examples, and partially classifying such actions, demonstrating their existence on all Damek-Ricci spaces.

Contribution

It offers new criteria and classifications for polar actions specifically on Damek-Ricci spaces, expanding understanding of their geometric symmetries.

Findings

Non-trivial polar actions exist on all Damek-Ricci spaces

Criteria for polar actions on Damek-Ricci spaces are established

Examples and partial classifications of polar actions are provided

Abstract

A proper isometric Lie group action on a Riemannian manifold is called polar if there exists a closed connected submanifold which meets all orbits orthogonally. In this article we study polar actions on Damek-Ricci spaces. We prove criteria for isometric actions on Damek-Ricci spaces to be polar, find examples and give some partial classifications of polar actions on Damek-Ricci spaces. In particular, we show that non-trivial polar actions exist on all Damek-Ricci spaces.